Would a TEE with a electric gate valve with a temperature control connected to Cold do this function? Can a gate valve be controlled by an aquastat??

: Does anyone have any advice?

No to the aquastat and gate valve.

You need a butterfly valve or ball valve. Why, with these electric valves you have limit switches built in and you have many, many models to choose from.There are makes of electric gate valves and electric sluice gate valves that use torque as the controller. These are used in much larger applications than what you are considering.